Laser-Induced Fano Resonance in Condensed Matter Physics

Abstract

Recent development of laser technology toward the realization of high-power laser has opened up a new research area exploring various fascinating phenomena governed by strongly photoexcited electronic states in diverse fields of science. In this chapter, we review the laser-induced Fano resonance (FR) in condensed matter systems, which is one of the representative resonance effects successfully exposed by strong laser field. The FR of concern sharply differs from FR effects commonly observed in conventional quantum systems where FR is caused by a weak external perturbation in a stationary system in the following two aspects. One is that the present FR is a transient phenomenon caused by nonequilibrium photoexcited states. The other is that this is induced by an optically nonlinear process. Here, we introduce two physical processes causing such transient and optically nonlinear FR in condensed matter, followed by highlighting anomalous effects inherent in it. The first is a Floquet exciton realized in semiconductor superlattices driven by a strong continuous-wave laser, and the second is the coherent phonon induced by an ultrashort pulse laser in bulk crystals.

1. Introduction

In quantum systems where discrete levels are embedded in energetically degenerate continuum states, resonance phenomenon is likely manifested, that is, characteristic of asymmetric spectral profiles consisting of both a peak and a dip. This is known as Fano resonance (FR) [1]; this is also termed as either Feshbach resonance or many-channel resonance. FR is one of the common and fundamental concepts in diverse fields of physics and chemistry; FR processes are observed, for instance, in strongly interacting Bose-Einstein condensates in an ultracold atomic system [2–4], superexcited states of molecules [5], a semiconductor quantum dot in an Aharonov-Bohm interferometer [6], an electronic transition near Weyl points strongly coupled with an infrared-active phonon in a Weyl semimetal [7].

In particular, within the restriction just to the FR processes triggered by laser irradiation, these may be classified in terms of the three categories as shown in Table 1. The first category is regarding whether a process is a linear one or a nonlinear one with respect to an order of a laser-matter interaction, as categorized as (a1) and (a2), respectively. For instance, the former is a photoabsorption process [8–11], and the latter is a multiphoton process [12–16]. The second category is regarding whether the process results from a built-in interaction between the discrete level and continuum that is intrinsic to a material itself or from a coupling induced extrinsically by a laser, as categorized as (b1) and (b2), respectively. For instance, the former is the interaction of an electron with a longitudinal optical (LO)-phonon in incoherent Raman scattering [17–22], and the electron-electron interaction brings about autoionization and the Auger process [23]. The latter FR process is known as a laser-induced continuum structure [2–4, 24]. The third category is regarding whether the process is a (quasi)stationary one or a transient one, as categorized as (c1) and (c2), respectively. In other words, this is whether (quasi)time-independent or time-dependent. For instance, the former is induced by a continuous-wave (cw) laser (monochromatic laser) [15, 16, 25, 26], and the latter is by a short pulsed laser [27–31].

Category	Characteristic
Optical process	(a1)	Linear (perturbative)	(a2)	Nonlinear (nonperturbative)
Interaction causing FR	(b1)	Intrinsic (built-in)	(b2)	Extrinsic (external)
Light source	(c1)	Monochromatic, continuous wave (stationary/quasistationary)	(c2)	Pulsed (transient)

Table 1.

Classification of FR into three categories.

It is stressed that for the FR categorized as (a2), its physical characters—such as asymmetry in spectral profile, spectral intensity, resonance position, and spectral width—are controllable in a quantum-mechanic manner by tuning various laser parameters. Thus, it is expected that underlying physics is enriched by intriguing effects inherent in this sort of FR. This differs from most of FR processes observed thus far because of being simply classified as (a1)-(b1)-(c1).

Currently, new research areas have been opened up owing to the progress of laser technology toward the realization of sophisticated high-power light sources. In particular, in the field of condensed matter physics, the development of high-intensity terahertz (THz) wave enables us to explore a photodressed quantum state in which a temporally periodic interaction of THz wave with matter is renormalized in the original quantum state in a nonperturbative manner [32–34]. Such an anomalous state is termed as a Floquet state because of ensuring the Floquet theorem [35]. Further, the development of ultrashort pulse laser—with its temporal width being of an order of 10 femtosecond (fs)—enables us to explore ultrafast transitory phenomena governed by strongly photoexcited electronic states. Bearing in mind such current situations, here, we focus exclusively on the laser-induced FR effects realized in the following two physical systems. One is a Floquet exciton formed in semiconductor superlattices (SLs) driven by a strong THz wave, and the other is a coherent phonon (CP) generated by ultrashort pulse laser in bulk crystals. In the light of Table 1, the FR effects of concern sharply differ from those commonly observed in conventional quantum systems classified as (a1)-(b1)-(c1) in the following aspects. Both of the Floquet exciton and the CP are induced by optically nonlinear processes, and hence the significant quantum controls of FR are feasible by means of tuning the respective applied light sources. Further, the Floquet exciton forms manifolds of quasistationary states with quasienergy as a constant of motion, where the FR is mediated by the ac-Zener tunneling caused by the THz wave. Hence, this is classified as (a2)-(b2)-(c1) and is herein termed as dynamic FR (DFR). On the other hand, the CP is a transient phenomenon caused by the built-in interaction of an LO-phonon with nonequilibrium photoexcited carriers. Hence, this is classified as (a2)-(b1)-(c2) and is herein termed as transient FR (TFR).

Below, we survey the present research backgrounds of DFR and TFR in brief. In both cases, the applied electric field of pumping laser is represented as F(t) = F₀(t) cos(ωt) with an envelope function F₀(t) at time t and the center frequency ω.

To begin with the DFR, this is closely related with the photodressed miniband formation [36]. Here, the cw laser with a constant amplitude F₀(t) ≡ F_ac gives rise to a nonlinear optical interaction with electron to result in a photodressed miniband with effective width Δ_eff = Δ₀∣J₀(x)∣, where Δ₀ and J₀(x) represent the width of the original SL miniband and the zeroth-order Bessel function of the first kind with x = eF_acd/ℏω, respectively, and e, d, and ℏ represent the elementary charge, a lattice constant of the SLs, and Planck’s constant divided by 2π, respectively. Each photodressed miniband forms a sequence of photon sidebands arrayed at equidistant energy intervals of ℏω following the Floquet theorem. The DFR is caused by the interaction due to the ac-Zener tunneling between photon sidebands pertaining to different sequences, and this is coherently controlled by tuning F_ac and ω. In particular, it is expected that an anomalous effect attributed to dynamic localization (DL) on DFR is revealed on the occasion that all of the photodressed minibands collapse by tuning x to ensure J₀(x) = 0 [36–38]. The DL was first observed in electron-doped semiconductor SLs driven by a THz wave [39]. In addition, this has also been observed in diverse physical systems such as a cold atomic gas in one-dimensional optical lattices [40], a Bose-Einstein condensate [41], and light in curved waveguide arrays [42–44].

As regards the TFR, this was observed in a lightly n-doped Si crystal immediately after carriers were excited by an ultrashort laser pulse [45], where the speculation was made that the observed FR would show the evidence of the birth of a polaronic-quasiparticle (PQ) likely formed in a strongly interacting carrier-LO-phonon system in a moment [46]. The TFR of concern has been observed exclusively in this system and semimetals/metals such as Bi and Zn [47, 48] till now, however, not observed in p-doped Si and GaAs crystals [49, 50]. Thus far, there are a number of theoretical studies regarding these experimental findings. The time-dependent Schrödinger equation in the system of GaAs was calculated to show the asymmetric shape featuring FR spectra, though apparently opposed to existing experimental results, as mentioned above [51]. Further, the classical Fano oscillator model was presented based on the Fano-Anderson Hamiltonian [52, 53], and the close comparison of the experimental results of the CP signals of Bi was made with the time signal obtained by taking the Fourier transform of Fano’s spectral formula into a temporal region [48]. Recently, the authors have constructed a fully quantum-mechanical model based on the PQ picture in a unified manner on an equal footing between both of polar and nonpolar semiconductors such as undoped GaAs and undoped Si [31]. Here, it has been shown that the TFR is manifested in a flash only before the carrier relaxation time (∼100 fs) in undoped Si, whereas this is absent from GaAs.

Acronyms used in the text and the corresponding meanings are summarized in Table 2. The remainder of this chapter is organized as follows. In Section 2, the theoretical framework is described, where the models of the DFR and TFR are presented separately in Sections 2.1 and 2.2, respectively. The results and discussion are given in Section 3, and the conclusion with summary is given in Section 4. Atomic units (a.u.) are used throughout unless otherwise stated.

Acronyms	Meanings
CP	Coherent phonon
cw	Continuous wave
DFR	Dynamic FR
DL	Dynamic localization
FR	Fano resonance
fs	Femtosecond
LO	Longitudinal optical
PQ	Polaronic-quasiparticle
SL	Superlattice
TFR	Transient FR
THz	Terahertz

Table 2.

Summary of acronyms used in text in alphabetical order and corresponding meanings.

2. Theoretical framework

2.1. Theoretical model for DFR in the photodressed exciton

2.1.1. Optical absorption spectra

The total Hamiltonian ${\widehat{H}}^{\left(\mathit{DFR}\right)}\left(t\right)$ concerned comprises a SL Hamiltonian consisting of field-free Hamiltonians of the conduction (c) and valence (v) bands, a Coulomb interaction between electrons, an intersubband interaction caused by the driving laser F(t) polarized in the direction of crystal growth (the z-axis), and an interband interaction caused by the probe laser f(t) = f_p cos(ω_pt) with the center frequency ω_p and the constant amplitude f_p; it is assumed that F_ac ≫ f_p and ω ≪ ω_p. The microscopic polarization defined as $p_{{\mathrm{\lambda \lambda }}^{\text{'}}{\mathit{k}}_{\parallel }}\left(t\right)\equiv ⟨a_{\mathrm{\lambda }{\mathit{k}}_{\parallel }}^{\left(v\right)†}{a}_{{\mathrm{\lambda }}^{\text{'}}{\mathit{k}}_{\parallel }}^{\left(c\right)}⟩$ is examined to shed light on the detail of DFR of the Floquet exciton; ⟨O⟩ represents the expectation value of the operator O. Here, λ⁽′⁾ = (b⁽′⁾, l⁽′⁾), which represents the lump of the SL miniband index b⁽′⁾ and the SL site l⁽′⁾. In addition, k_∥ represents the in-plane momentum associated with the relative motion of the pair of c band and v band electrons, where the in-plane is defined as the plane normal to the z-axis; hereafter, the relative position conjugate to k_∥ is represented as ρ. Further, $a_{\mathrm{\lambda }{\mathit{k}}_{\parallel }}^{\left(s\right)†}\left(a_{\mathrm{\lambda }{\mathit{k}}_{\parallel }}^{\left(s\right)}\right)$ represents the creation (annihilation) operator of the electron with λ and k_∥ in band s.

The equation of motion for the microscopic polarization is given by the semiconductor Bloch equation

$i\left(\frac{d}{\mathit{dt}}+\frac{1}{T_2}\right)p_{\mathrm{\lambda }{\mathrm{\lambda }}^{\prime }{\mathit{k}}_{\parallel }}\left(t\right)=⟨[a_{\mathrm{\lambda }{\mathit{k}}_{\parallel }}^{\left(v\right)†}{a}_{\mathrm{\lambda }{\mathit{k}}_{\parallel }}^{\left(c\right)},{\widehat{H}}^{\left(\mathit{DFR}\right)}\left(t\right)]⟩$E1

with T₂ dephasing time. For the practical purpose of tackling the multichannel scattering problem of exciton, it is convenient to transform it into the equation for $\overline{p}\left(\mathit{\rho },z_v,z_c,t\right)$ defined in the real-space representation as

$\overline{p}\left(\mathit{\rho },z_v,z_c,t\right)=e^{i{\omega }_{p}t}\sum _{\mathrm{\lambda },{\mathrm{\lambda }}^{\prime }}\int d{\mathit{k}}_{\parallel }{e}^{i{\mathit{k}}_{\parallel }\cdot \mathit{\rho }}⟨z_v|\mathrm{\lambda }⟩p_{\mathrm{\lambda }{\mathrm{\lambda }}^{\prime }{\mathit{k}}_{\parallel }}\left(t\right)⟨{\mathrm{\lambda }}^{\prime }|z_c⟩,$E2

where ⟨λ| z⟩ represents the Wannier function at position z − ld in SL miniband b. The resulting equation becomes

$\begin{array}{l}i\left(\frac{d}{\mathit{dt}}+\frac{1}{T_2}-i{\omega }_p\right)\overline{p}\left(\mathit{\rho },z_v,z_c,t\right)+{\left(2\pi \right)}^2{e}^{i{\omega }_{p}t}{f}_0^{\left(+\right)}\left(t\right)d_0^{\left(\mathit{vc}\right)}\delta \left(\mathit{\rho }\right)\delta \left(z_v-z_c\right)\hfill \\ =\int \mathit{dz}\left[\overline{p}\left(\mathit{\rho },z_v,z,t\right)H_{\mathrm{TB}}^{\left(c\right)}\left(z,z_c,t\right)-H_{\mathrm{TB}}^{\left(v\right)}\left(z_v,z,t\right)\overline{p}\left(\mathit{\rho },z,z_c,t\right)\right]+\mathcal{H}\left(\mathit{\rho },z_v,z_c\right)\overline{p}\left(\mathit{\rho },z_v,z_c,t\right),\hfill \end{array}$E3

where the rotating wave approximation is employed by replacing f(t) by $f_0^{\left(+\right)}\left(t\right)\equiv \left(f_p/2\right)e^{-i{\omega }_{p}t}$ and $d_0^{\left(\mathit{vc}\right)}$ represents the interband dipole moment of a bulk material. Here, the Hamiltonian $\mathcal{H}$(ρ, z_v, z_c) for the in-plane motion is given by $\mathcal{H}\left(\mathit{\rho },z_v,z_c\right)=-{\text{∇}}_{\mathit{\rho }}^2/2m_{\parallel }+V\left(r\right)$, where m_∥ and V(r) =  − 1/(ε₀r) represent an in-plane reduced mass and the Coulomb interaction, respectively, with $r=\sqrt{{\mathit{\rho }}^2+{\left(z_v-z_c\right)}^2}$ and ε₀ the dielectricity of vacuum. The nearest-neighbor tight-binding Hamiltonian of the laser-driven SLs is given by $H_{\mathrm{TB}}^{\left(s\right)}\left(z,z^{\prime },t\right)\equiv ⟨z|{\widehat{H}}_{\mathrm{TB}}^{\left(s\right)}\left(t\right)|z^{\prime }⟩$, where

${\widehat{H}}_{\mathrm{TB}}^{\left(s\right)}\left(t\right)=\sum _{\mathrm{\lambda }=\left(b,l\right)}\left[{\epsilon }_{0b}^{\left(s\right)}|\mathrm{\lambda }⟩⟨\mathrm{\lambda }|+\frac{{\left(-1\right)}^{b+{\sigma }^s}}{4}{\Delta }_b^{\left(s\right)}\left(|l,b⟩⟨l+1,b|+|l+1,b⟩⟨l,b|\right)\right]-F\left(t\right)\sum _{\mathrm{\lambda }{\mathrm{\lambda }}^{\prime }}\mid \mathrm{\lambda }⟩Z_{\mathrm{\lambda }{\mathrm{\lambda }}^{\prime }}^{\left(s\right)}⟨{\mathrm{\lambda }}^{\prime }\mid ,$E4

and ${\epsilon }_{0b}^{\left(s\right)}$ and ${\Delta }_b^{\left(s\right)}$ represent the band center and the band width of b, respectively, with σ^s = 0 (for s = c) and 1 (for s = v). The last term of Eq. (4) represents the dipole interaction induced by the driving laser F(t) with $Z_{\mathrm{\lambda },{\mathrm{\lambda }}^{\prime }}^{\left(s\right)}$ as a dipole matrix element. It should be noted that the off-diagonal contribution of $Z_{\mathrm{\lambda },{\mathrm{\lambda }}^{\prime }}^{\left(s\right)}$ with b ≠ b′ induces the ac-Zener tunneling, which plays a significant role of quantum control of DFR, as shown below.

The concerned function $\overline{p}\left(\mathit{\rho },z_v,z_c,t\right)$ can be expressed in terms of the complete set of the Floquet wave functions {ψ_Eβ(ρ, z_v, z_c, t)}, that is,

$\overline{p}\left(\mathit{\rho },z_v,z_c,t\right)=\int \mathit{dE}\sum _{\beta }{a}_{\mathit{E\beta }}{\psi }_{\mathit{E\beta }}\left(\mathit{\rho },z_v,z_c,t\right)$E5

with a_Eβ as an expansion coefficient. Here, the Floquet wave function ensures the following homogeneous equation associated with the inhomogeneous equation of Eq. (3) as

$\begin{array}{l}i\left(\frac{d}{\mathit{dt}}+E\right){\psi }_{\mathit{E\beta }}\left(\mathit{\rho },z_v,z_c,t\right)=\int \mathit{dz}\left[{\psi }_{\mathit{E\beta }}\left(\mathit{\rho },z_v,z,t\right)H_{\mathrm{TB}}^{\left(c\right)}\left(z,z_c,t\right)-H_{\mathrm{TB}}^{\left(v\right)}\left(z_v,z,t\right){\psi }_{\mathit{E\beta }}\mathit{\rho },z,z_c,t\right)]\hfill \\ +\mathcal{H}\left(\mathit{\rho },z_v,z_c\right){\psi }_{\mathit{E\beta }}\left(\mathit{\rho },z_v,z_c,t\right),\hfill \end{array}$E6

where the temporally periodic boundary condition ψ_Eβ(ρ, z_v, z_c, t) = ψ_Eβ(ρ, z_v, z_c, t + T) is imposed on it with E and T = 2π/ω as quasienergy and the time period of the driving laser field, respectively. Equation (6) is the Wannier equation of the Floquet exciton of concern. It should be noted that this is cast into the multichannel scattering equations and the Floquet state of ψ_Eβ(ρ, z_v, z_c, t) forms a continuum spectrum designated by both E and β with β representing the label of an open channel. Such a multichannel feature is introduced by the strong driving laser F(t) that closely couples an excitonic-bound state with continua; more detail of the multichannel scattering problem is described in Section 2.1.2. The expansion coefficient a_Eβ is readily obtained by inserting Eq. (5) into Eq. (3) in view of Eq. (6) as

$a_{\mathit{E\beta }}=\frac{{\left(2\pi \right)}^2{d}_0^{\left(\mathit{vc}\right)}\left(f_p/2\right)}{\left(E-{\omega }_p-\mathit{i\gamma }\right)T}{\int }_0^{T}d{t}^{\prime }{\overline{\psi }}_{\mathit{E\beta }}\left(t^{\prime }\right),$E7

where ${\overline{\psi }}_{\mathit{E\beta }}\left(t\right)=\int \mathit{dz}{\psi }_{\mathit{E\beta }}\left(\mathbf{0},z,z,t\right)$ and γ = 1/T₂.

Since the macroscopic polarization is given by $P\left(t\right)={\sum }_{\mathrm{\lambda },{\mathrm{\lambda }}^{\prime }}\int d{\mathit{k}}_{\parallel }{d}_0^{\left(\mathit{vc}\right)\ast }{p}_{\mathrm{\lambda }{\mathrm{\lambda }}^{\prime }{\mathit{k}}_{\parallel }}\left(t\right)$, the linear optical susceptibility χ(t) with respect to the weak probe laser f(t) is cast into [54]

$\chi \left(t\right)=\frac{{\left|d_0^{\left(\mathit{vc}\right)}\right|}^2}{{\epsilon }_0}\int \mathit{dE}\sum _{\beta }\frac{O_{\mathit{E\beta }}\left(t\right)}{E-{\omega }_p-\mathit{i\gamma }},$E8

where $O_{\mathit{E\beta }}\left(t\right)=\left[{\overline{\psi }}_{\mathit{E\beta }}\left(t\right)/T\right]{\int }_0^{T}d{t}^{\prime }{\overline{\psi }}_{\mathit{E\beta }}^{\ast }\left(t^{\prime }\right)$. Taking the Fourier transform of χ(t) ≡ ∑_je^ijωtχ_j(ω_p; ω), leads to the expression of the absorption coefficient to be calculated as

$\alpha \left({\omega }_p;\omega \right)=\frac{{\omega }_p}{C}\sum _{j}Im{\chi }_j\left({\omega }_p;\omega \right)$E9

with C the speed of light; ${\chi }_{j\cancel{=}0}\left({\omega }_p;\omega \right)$ vanishes in the limit of F_ac→0.

2.1.2. Multichannel scattering problem

The absorption coefficient of Eq. (9) is obtained by evaluating a set of the wave functions, {ψ_Eβ(ρ, z_v, z_c, t)}. To do this, first, the wave function is expanded as

${\psi }_{\mathit{E\beta }}\left(\mathit{\rho },z_v,z_c,t\right)=\sum _{\mu }{\mathrm{\Phi }}_{\mu }\left(z_v,z_c,t\right)F_{\mathit{\mu \beta }}\left(\rho \right),$E10

where ρ = ∣ρ∣, and just the contribution of the s-angular-momentum component is incorporated because of little effects from higher-order components. Here, Φ_μ(z_v, z_c, t) is the real-space representation of the Floquet state ∣Φ_μ⟩, that is, Φ_μ(z_v, z_c, t) = ⟨z_v, z_c| Φ_μ⟩, satisfying $\left({\widehat{H}}_{\mathrm{TB}}-i\text{∂}/\text{∂}t\right)\mid {\Phi }_{\mu }⟩={\mathcal{E}}_{\mu }\mid {\Phi }_{\beta }⟩,$ where ${\widehat{H}}_{\mathrm{TB}}\equiv {\widehat{H}}_{\mathrm{TB}}^{\left(c\right)}+{\widehat{H}}_{\mathrm{TB}}^{\left(v\right)}$ and $\mathcal{E}$_μ is the μth quasienergy. The index μ is considered as the approximate quantum number $\mu \approx \left[\overline{\mu },k\right]$ with $\overline{\mu }\equiv \left[b_c,b_v,n_p\right]$ as a photon sideband index, where b_c and b_v are SL miniband indexes belonging to the c- and v-bands, respectively, and k and n_p represent the Bloch momentum of the joint miniband of (b_c, b_v) and the number of photons relevant to absorption and emission, respectively. The quantum number $\overline{\mu }$ becomes a set of the good quantum numbers with F_ac decreasing, while k always remains conserved because of the spatial periodicity in the laser-driven SLs of concern. In view of Eq. (10), Eq. (6) is recast into the coupled equations for the radial wave function F_νβ(ρ), that is,

$\sum _{\mu }{L}_{\mathit{\mu \nu }}{F}_{\mathit{\nu \beta }}\left(\rho \right)={\mathit{EF}}_{\mathit{\mu \beta }}\left(\rho \right),$E11

where L_μν is an operator given by L_μν = δ_μν[−(2m_∥)¹(d²/dρ² + ρ¹d/dρ) + $\mathcal{E}$_μ] + V_μν(ρ) and V_μν(ρ) is a Coulomb matrix element defined as $V_{\mathit{\mu \nu }}\left(\rho \right)=T^{-1}{\int }_0^T\mathit{dt}\int {\mathit{dz}}_v\int {\mathit{dz}}_c{\Phi }_{\mu }^{\ast }\left(z_v,z_c,t\right)V\left(\rho ,z_v,z_c\right){\Phi }_{\nu }\left(z_v,z_c,t\right).$

The Floquet exciton in the laser-driven SL system pertains to the multichannel scattering problem, because V_μν(ρ) vanishes at ρ ≫ 1. Actually, for a given E, the channel μ satisfying E > $\mathcal{E}$_μ is an open channel, while the channel μ satisfying E < $\mathcal{E}$_μ is a closed channel. Thus, the label μ of F_μβ plays the role of the scattering channel. On the other hand, the label β means the number of independent solutions satisfying Eq. (11). Here, there are same number of independent solutions as open channels, since as many scattering boundary conditions are imposed on F_μβ at ρ ≫ 1; while evanescent boundary conditions that F_μβ vanishes at ρ ≫ 1 are imposed on closed channels. Eq. (11) can be numerically evaluated by virtue of the R-matrix propagation method, which is a sophisticated formalism providing a stable numerical algorithm with extremely high accuracy [55].

It is expected that the DFR of concern is caused by a coupling between photon sidebands mediated by ac-Zener tunneling, as mentioned in Section 1. To see this situation in more detail, Figure 1 shows the interacting two photon sidebands $\overline{\mu }$ and ${\overline{\mu }}^{\prime }\equiv \left[b_c^{\prime },b_v^{\prime },n_p^{\prime }\right]$, where the discrete Floquet excitonic state is supported by the photon sideband $\overline{\mu }$, and this is also embedded in the continuum of the alternative photon sideband ${\overline{\mu }}^{\prime }$. It is likely that the DFR occurs due to a close coupling between these photon sidebands, and, eventually, the exciton state decays into the continuum state pertaining to ${\overline{\mu }}^{\prime }$. In fact, it is noted that the Coulomb interaction incorporated in Eq. (6) also gives rise to FR. Defining the difference between the photon numbers of both photon sidebands, namely, $\Delta n_p=\mid n_p-n_p^{\prime }\mid$, the ac-Zener tunneling is featured by Δn_p ≠ 0, while the Coulomb coupling is by Δn_p = 0. The spectral profile and intensity of FR in the former can be even more effectively controlled than in the latter by modulating the laser parameters F_ac and ω, since the degree of magnitude of ac-Zener tunneling depends exclusively on both of the external parameters, differing from the Coulomb interaction. In the region of F_ac weak enough to suppress the ac-Zener tunneling, the FR is dominated by the Coulomb coupling, similarly to the conventional FR observed in the original SLs without laser irradiation [56].

Figure 1.

Schematic diagram of the DFR formation in the Floquet excitonic system. This shows the coupling mechanism that a bound state supported by the sideband $\overline{\mu }$ interacts with a continuum state belonging to the sideband ${\overline{\mu }}^{\prime }$ by the ac-Zener tunneling to result in the Fano decay (from Ref. [15] with partial modification).

2.2. Theoretical model for TFR in the CP generation

2.2.1. Introduction of polaronic quasiparticle operators

The total Hamiltonian ${\widehat{H}}^{\left(\mathit{TFR}\right)}$ of concern is given by ${\widehat{H}}^{\left(\mathit{TFR}\right)}={\widehat{H}}_e+\widehat{H}\prime \left(t\right)+{\widehat{H}}_p+{\widehat{H}}_{e-p}.$ Here, ${\widehat{H}}_e$ represents an electron Hamiltonian including an interelectronic Coulomb potential, where a two-band model is employed that consists of the energetically lowest c band and the energetically highest valence v band, and a creation (annihilation) operator of electron with band index b and Bloch momentum k is represented as $a_{b\mathit{k}}^{†}\left(a_{b\mathit{k}}\right)$. ${\widehat{H}}_p$ represents an LO-phonon Hamiltonian, where a creation (annihilation) operator of LO-phonon with an energy dispersion ${\omega }_{\mathit{q}}^{\left(\mathit{LO}\right)}$ at momentum q is represented as $c_{\mathit{q}}^{†}\left(c_{\mathit{q}}\right)$. Further, ${\widehat{H}}^{\prime }\left(t\right)$ and ${\widehat{H}}_{e-p}$ represent interaction Hamiltonians of electron with the pump pulse and the LO-phonon, respectively. These are given as follows:

${\widehat{H}}^{\prime }\left(t\right)=-\frac{1}{2}\sum _{b,b^{\prime }\left(\cancel{=}b\right),\mathit{k}}\left[{\Omega }_{b{b}^{\prime }}\left(t\right)a_{b\mathit{k}}^{†}{a}_{b^{\prime }\mathit{k}}+{\Omega }_{b{b}^{\prime }}^{\ast }\left(t\right)a_{b^{\prime }\mathit{k}}^{†}{a}_{b\mathit{k}}\right],$E12

where Ω_bb^′(t) = d_bb^′F(t) with d_bb^′ an electric dipole moment between b and b^′ bands, and

${\widehat{H}}_{e-p}=\sum _{b,\mathit{q},\mathit{k}}\left(g_{b\mathit{q}}{c}_{\mathit{q}}{a}_{b\mathit{k}+\mathit{q}}^{†}{a}_{b\mathit{k}}+g_{b\mathit{q}}^{\ast }{c}_{\mathit{q}}^{†}{a}_{b\mathit{k}}^{†}{a}_{b\mathit{k}+\mathit{q}}\right),$E13

where g_bq is a coupling constant of b band electron with the LO-phonon. Here, let the envelope of F(t) be of squared shape just for the sake of simplicity, that is, F₀(t) = $\mathcal{F}$₀θ(t + τ_L/2)θ(τ_L/2 − t) with $\mathcal{F}$₀ constant, where temporal width τ_L is of the order of a couple of 10 fs at most, satisfying ${\tau }_L\ll 2\pi /{\omega }_{\mathit{q}}^{\left(\mathit{LO}\right)}$.

The equation of motion of a composite operator $A_{\mathit{q}}^{†}\left(\mathit{k}b{b}^{\prime }\right)\equiv a_{b,\mathit{k}+\mathit{q}}^{†}{a}_{b^{\prime }\mathit{k}}$ is considered below, where this represents an induced carrier density with spatial anisotropy determined by q; ∣q∣ is finite, though quite small, that is, q ≠ 0. It is convenient to remove from this equation high-frequency contributions by means of the rotating-wave approximation [57] by replacing $A_{\mathit{q}}^{†}\left(\mathit{k}b{b}^{\prime }\right)$ by $e^{i{\overline{\omega }}_{b{b}^{\prime }}t}{\overline{A}}_{\mathit{q}}^{†}\left(\mathit{k}b{b}^{\prime }\right)$, where ${\overline{\omega }}_{\mathit{cv}}=\omega$, ${\overline{\omega }}_{\mathit{vc}}=-\omega$, and ${\overline{\omega }}_{\mathit{bb}}=0$. Thus, the resulting equation of motion is as follows:

$\begin{array}{lll}-i\left(\frac{d}{\mathit{dt}}+\frac{1}{T_{\mathit{q}}\left(\mathit{k}b{b}^{\prime }\right)}\right){{\displaystyle \overline{A}}}_{\mathit{q}}^{†}\left(\mathit{k}b{b}^{\prime }\right)\hfill & =\hfill & \left[{{\displaystyle \widehat{\mathcal{H}}}}_e\left(t\right),{{\displaystyle \overline{A}}}_{\mathit{q}}^{†}\left(\mathit{k}b{b}^{\prime }\right)\right]-{{\displaystyle \overline{A}}}_{\mathit{q}}^{†}\left(\mathit{k}b{b}^{\prime }\right){\overline{\omega }}_{b{b}^{\prime }}+\left[{\widehat{H}}_{e-p},{\overline{A}}_{\mathit{q}}^{†}\left(\mathit{k}b{b}^{\prime }\right)\right]\hfill \\ \hfill & \approx \hfill & \sum _{{\displaystyle \tilde{\mathit{k}}}{\displaystyle \tilde{b}}{{\displaystyle \tilde{b}}}^{\prime }}{{\displaystyle \overline{A}}}_{\mathit{q}}^{†}\left({\displaystyle \tilde{\mathit{k}}}{\displaystyle \tilde{b}}{{\displaystyle \tilde{b}}}^{\prime }\right){{\displaystyle \overline{Z}}}_{\mathit{q}}\left({\displaystyle \tilde{\mathit{k}}}\tilde{b}{\tilde{b}}^{\prime },\mathit{k}b{b}^{\prime }\right)+\left[{\widehat{H}}_{e-p},{{\displaystyle \overline{A}}}_{\mathit{q}}^{†}\left(\mathit{k}b{b}^{\prime }\right)\right],\hfill \end{array}$E14

where the total electronic Hamiltonian is defined as ${\widehat{\mathcal{H}}}_e\left(t\right)={\widehat{H}}_e+{\widehat{H}}^{\prime }\left(t\right),$ the first commutator in the right-hand side of the first equality is evaluated by making a factorization approximation, and T_q(kbb^′) represents a phenomenological relaxation time constant relevant to $A_{\mathit{q}}^{†}\left(\mathit{k}b{b}^{\prime }\right)$. Further, ${\overline{Z}}_{\mathit{q}}$ represents a non-Hermitian matrix, which is a slowly varying function in time, since rapidly time-varying contributions are removed owing to the above rotating-wave approximation, aside from the discontinuity at t =  ± τ_L/2.

Bearing in mind this situation, we tackle left and right eigenvalue problems of ${\overline{Z}}_{\mathit{q}}$ [58], described by $U_{\mathit{q}}^{L†}{\overline{Z}}_{\mathit{q}}={\mathcal{E}}_{\mathit{q}}{U}_{\mathit{q}}^{L†}$ and ${\overline{Z}}_{\mathit{q}}{U}_{\mathit{q}}^R=U_{\mathit{q}}^R{\mathcal{E}}_{\mathit{q}},$ respectively, in terms of an adiabatic-eigenvalue diagonal matrix $\mathcal{E}$_q and the associated biorthogonal set of adiabatic eigenvectors $\left\{U_{\mathit{q}}^L,U_{\mathit{q}}^R\right\}$ with time t fixed as a parameter. The orthogonality relation and the completeness are read as $U_{\mathit{q}}^{L†}{U}_{\mathit{q}}^R=1$ and $U_{\mathit{q}}^R{U}_{\mathit{q}}^{L†}=1$, respectively [58]. Given the relation ${\overline{Z}}_{\mathit{q}}=U_{\mathit{q}}^R{\mathcal{E}}_{\mathit{q}}{U}_{\mathit{q}}^{L†}$, Eq. (14) is recast into the form of adiabatic coupled equations:

$-i\frac{{\mathit{dB}}_{\mathit{q}\alpha }^{†}}{\mathit{dt}}=B_{\mathit{q}\alpha }^{†}{\mathcal{E}}_{\mathit{q}\alpha }+i\sum _{{\alpha }^{\prime }}{B}_{\mathit{q}{\alpha }^{\prime }}^{†}{\mathcal{W}}_{\mathit{q}{\alpha }^{\prime }\alpha }+\left[{\widehat{H}}_{e-p},B_{\mathit{q}\alpha }^{†}\right],$E15

where the operator $B_{\mathit{q}\alpha }^{†}$ is defined as $B_{\mathit{q}\alpha }^{†}\equiv {\overline{A}}_{\mathit{q}}^{†}{U}_{\mathit{q}\alpha }^R$, ${\mathcal{W}}_{\mathit{q}{\alpha }^{\prime }\alpha }\equiv \left[{\mathit{dU}}_{\mathit{q}{\alpha }^{\prime }}^{L†}/\mathit{dt}\right]U_{\mathit{q}\alpha }^R,+U_{\mathit{q}{\alpha }^{\prime }}^{L†}{T}_{\mathit{q}}^{-1}{U}_{\mathit{q}\alpha }^R$, and $\mathcal{E}$_qα(t) is complex adiabatic energy at time t associated with the operator $B_{\mathit{q}\alpha }^{†}\left(t\right)$ thus introduced. Hereafter, this operator is termed as a creation operator of quasiboson, and the corresponding annihilation operator is defined as $B_{\mathit{q}\alpha }\equiv U_{\mathit{q}\alpha }^{R†}{\overline{A}}_{\mathit{q}}.$ The set of eigenstates {α} is composed of continuum states represented as β with eigenenergy $\mathcal{E}$_qβ and a single discrete energy state represented as α₁ with eigenenergy $\mathcal{E}$_qα1, that is, {α} = ({β}, α₁); the state β corresponds to electron-hole continuum arising from interband transitions, and the state α₁ corresponds to a plasmon-like mode. It is noted that the relation of $⟨\left[B_{\mathit{q}\alpha }\left(t\right),B_{{\mathit{q}}^{\prime }{\alpha }^{\prime }}^{†}\left(t\right)\right]⟩={\delta }_{\mathit{q}{\mathit{q}}^{\prime }}{\delta }_{\alpha {\alpha }^{\prime }}$ is assumed, though B_qα(t) and $B_{\mathit{q}\alpha }^{†}\left(t\right)$ do not satisfy the equal-time commutation relations for a real boson, where $〈\widehat{X}〉$ means an expectation value of operator $\widehat{X}$ with respect to the ground state; the validity of the criterion of this relation is discussed in detail in Ref. [31].

Eq. (13) is rewritten as ${\widehat{H}}_{e-p}={\sum }_{\mathit{q},\alpha }\left(c_{\mathit{q}}{B}_{\mathit{q}\alpha }^{†}{M}_{\mathit{q}\alpha }+M_{\mathit{q}\alpha }^{\ast }{B}_{\mathit{q}\alpha }{c}_{\mathit{q}}^{†}\right)$ with an effective coupling between quasiboson and LO-phonon as $M_{\mathit{q}\alpha }={\sum }_{\mathit{k}b}{g}_{b\mathit{q}}{U}_{\mathit{q}\alpha }^{L†}\left(\mathit{k}\mathit{bb}\right)$. Thus, the commutator in Eq. (15) is approximately evaluated as $\left[{\widehat{H}}_{e-p},B_{\mathit{q}\alpha }^{†}\right]\approx M_{\mathit{q}\alpha }^{\text{'}\ast }{c}_{\mathit{q}}^{†}$, though $M_{\mathit{q}\alpha }^{\text{'}\ast }\ne M_{\mathit{q}\alpha }^{\ast }$. On the other hand, the equation of motion of the LO-phonon is described by $-{\mathit{idc}}_{\mathit{q}}^{†}/\mathit{dt}={\sum }_{\alpha }{B}_{\mathit{q}\alpha }^{†}{M}_{\mathit{q}\alpha }+c_{\mathit{q}}^{†}{\omega }_{\mathit{q}}^{\left(\mathit{LO}\right)}.$ Both of the equations of motion for $B_{\mathit{q}}^{†}$ and $c_{\mathit{q}}^{†}$ are integrated into a single equation in terms of matrix notations as follows:

$-i\frac{d}{\mathit{dt}}\left(B_{\mathit{q}}^{†},c_{\mathit{q}}^{†}\right)\approx \left(B_{\mathit{q}}^{†},c_{\mathit{q}}^{†}\right)h_{\mathit{q}}+\left({\mathit{iB}}_{\mathit{q}}^{†}{\mathcal{W}}_{\mathit{q}},0\right).$E16

Here, the non-Hermitian matrix h_q ≡ {h_qγγ^′} given by $h_{\mathit{q}}=\left(\begin{array}{ll}{\mathcal{E}}_{\mathit{q}}\hfill & M_{\mathit{q}}\hfill \\ {M^{\prime }}_{\mathit{q}}^{†}\hfill & {\omega }_{\mathit{q}}^{\left(\mathit{LO}\right)}\hfill \end{array}\right)$ is introduced with γ , γ^′ = 1 ∼ N + 2, where N represents the number of electron-hole (discretized) continua, namely, β = 1 ~ N, aside from two discrete states attributed to a plasmon-like mode and an LO-phonon mode designated by α₁ and α₂, respectively: {γ} = ({β}, α₁, α₂). In the system of the TFR of concern, the case is exclusively examined that both of the discrete levels of α₁ and α₂ are embedded into the continua {β}. Thus, the following coupled equations for the multichannel scattering problem are taken account of

$\sum _{{\gamma }^{\prime }}{h}_{\mathit{q}\gamma {\gamma }^{\prime }}{V}_{\mathit{q}{\gamma }^{\prime }\beta }^R=V_{\mathit{q}\mathit{\gamma \beta }}^R{\mathcal{E}}_{\mathit{q}\beta },$E17

where $V_{\mathit{q}\beta }^R=\left\{V_{\mathit{q}\mathit{\gamma \beta }}^R\right\}$ is the right vector representing the solution for given energy $\mathcal{E}$_qβ; similarly to Eq. (11) for the DFR, the indices of γ and β play the roles of a scattering channel and the number of independent solutions, respectively. Eq. (17) provides the theoretical basis on which both of LO-phonon and plasmon-like modes are brought into connection with the CP dynamics on an equal footing. In terms of this vector, a set of N-independent operators, $F_{\mathit{q}\beta }^{†}\left(\beta =1\sim N\right)$, is defined as

$F_{\mathit{q}\beta }^{†}=\sum _{{\beta }^{\prime }}{B}_{\mathit{q}{\beta }^{\prime }}^{†}{V}_{\mathit{q}{\beta }^{\prime }\beta }^R+B_{\mathit{q}{\alpha }_1}^{†}{V}_{\mathit{q}{\alpha }_1\beta }^R+c_{\mathit{q}}^{†}{V}_{\mathit{q}{\alpha }_2\beta }^R.$E18

In addition, the left vector $V_{\mathit{q}\beta }^{L†}=\left\{V_{\mathit{q}\mathit{\beta \gamma }}^{L†}\right\}$ associated with $V_{\mathit{q}\beta }^R$ is introduced to ensure the inverse relations $B_{\mathit{q}\alpha }^{†}=F_{\mathit{q}}^{†}{V}_{\mathit{q}\alpha }^{L†}$ and $c_{\mathit{q}}^{†}=F_{\mathit{q}}^{†}{V}_{\mathit{q}{\alpha }_2}^{L†},$ where $V_{\mathit{q}}^{L†}{V}_{\mathit{q}}^R=1$ and $V_{\mathit{q}}^R{V}_{\mathit{q}}^{L†}=1$. Hereafter, the operator $F_{\mathit{q}\beta }^{†}\left(t\right)$ thus introduced is termed as a creation operator of PQ, and then the corresponding annihilation operator is F_qβ(t); these are not bosonic operators. The bosonization scheme for the PQ operators is similar to that for the quasiboson operators, where the PQ ground state is given by the direct product of the ground states of quasiboson and LO-phonon and $\mathcal{E}$_qβ(t) is read as the single-PQ adiabatic energy at time t with mode qβ.

Given Eq. (18), Eq. (16) becomes adiabatic coupled equations for $F_{\mathit{q}}^{†}$:

$-i\frac{d}{\mathit{dt}}{F}_{\mathit{q}\beta }^{†}\approx F_{\mathit{q}\beta }^{†}{\mathcal{E}}_{\mathit{q}\beta }+i\sum _{{\beta }^{\prime }}{F}_{\mathit{q}{\beta }^{\prime }}^{†}{\mathcal{I}}_{\mathit{q}{\beta }^{\prime }\beta },$E19

where ${\mathcal{I}}_{\mathit{q}}=\left({\mathit{dV}}_{\mathit{q}}^{L†}/\mathit{dt}\right)V_{\mathit{q}}^R+V_{\mathit{q}}^{L†}{\mathcal{W}}_{\mathit{q}}{V}_{\mathit{q}}^R.$ In terms of F_q and $F_{\mathit{q}}^{†}$, the associated retarded Green function is given by [59]

$G_{\mathit{q}\beta {\beta }^{\prime }}^R\left(t,t^{\prime }\right)=-\mathit{i\theta }\left(t-t^{\prime }\right)⟨\left[F_{\mathit{q}\beta }\left(t\right),F_{\mathit{q}{\beta }^{\prime }}^{†}\left(t^{\prime }\right)\right]⟩.$E20

2.2.2. Transient induced photoemission spectra

A weak external potential f_q(t) additionally introduced in the transient and nonequilibrium system of concern induces an electron density $n_{\mathit{q}}^{\left(\mathit{ind}\right)}\left(t\right)$ given by

$n_{\mathit{q}}^{\left(\mathit{ind}\right)}\left(t\right)=\frac{1}{4\mathit{\pi V}}{\int }_{-\infty }^{t}d{t}^{\prime }{\chi }_{\mathit{q}}^{\left(t\right)}\left(t,t^{\prime }\right)f_{\mathit{q}}\left(t^{\prime }\right),$E21

based on the linear response theory [59, 60] with V the volume of crystal. It is noted that $n_{\mathit{q}}^{\left(\mathit{ind}\right)}\left(t\right)$ is nonlinear with respect to the pump field. Here, ${\chi }_{\mathit{q}}^{\left(t\right)}\left(t,t^{\prime }\right)$ represents the retarded longitudinal susceptibility that depends on passage of t and the relative time τ = t − t^′, differing from equilibrium systems depending solely on τ. Introducing a retarded longitudinal susceptibility due to the electron-induced interaction and that of an LO-phonon-induced interaction represented as χ_q(t, t^′) and ${\chi }_q^{\prime }\left(t,t^{\prime }\right)$, respectively, ${\chi }_{\mathit{q}}^{\left(t\right)}\left(t,t^{\prime }\right)$ is given by [59]

${\chi }_{\mathit{q}}^{\left(t\right)}\left(t,t^{\prime }\right)={\chi }_{\mathit{q}}\left(t,t^{\prime }\right)+{\chi }_{\mathit{q}}^{\prime }\left(t,t^{\prime }\right).$E22

Let f_q(t^′) be assumed to be f_q(t^′) = f_q0 δ(t^′ − t_p) in the present system; f_q0 is independent of t^′, and t_p represents the time at which f_q(t^′) probes transient dynamics of concern. Thus, it is seen that ${\chi }_{\mathit{q}}^{\left(t\right)}\left(t,t_p\right)$ reveals the way of alteration of $n_{\mathit{q}}^{\left(\mathit{ind}\right)}\left(t\right)$ after t_p, since Eq. (21) becomes $n_{\mathit{q}}^{\left(\mathit{ind}\right)}\left(t\right)=f_{\mathit{q}0}{\chi }_{\mathit{q}}^{\left(t\right)}\left(t,t_p\right)\theta \left(t-t_p\right)/4\mathit{\pi V}.$

In terms of ${\chi }_{\mathit{q}}^{\left(t\right)}\left(t_p+\tau ,t_p\right)$, the dielectric function ε_q(t_p + τ, t_p) is readily obtained, and by taking the Fourier transform of it as ${\tilde{\epsilon }}_{\mathit{q}}\left(t_p;{\omega }_p\right)={\int }_0^{\infty }\mathit{d\tau }{e}^{-i{\omega }_p\tau }{\epsilon }_{\mathit{q}}\left(t_p+\tau ,t_p\right)$, this leads to a transient absorption coefficient α_q(t_p; ω_p) at time t_p. This is given by α_q(t_p; ω_p) = ωA_q(t_p; ω_p)/n(t_p; ω_p)C, where $A_{\mathit{q}}\left(t_p;{\omega }_p\right)=Im{\tilde{\epsilon }}_{\mathit{q}}\left(t_p;{\omega }_p\right)$ and n(t_p; ω_p) represents the index of refraction. It is remarked that according to the definition of the sign of ω_p made above, transient photoemission spectra, where A_q(t_p; ω_p) < 0, peak at positive ω_p, while transient photoabsorption spectra, where A_q(t_p; ω_p) > 0, peak at negative ω_p. For the sake of the later convenience, the transient induced photoemission spectra are defined as ${\overline{A}}_{\mathit{q}}\left(t_p;{\omega }_p\right)=-A_{\mathit{q}}\left(t_p;{\omega }_p\right).$

Based on the PQ model developed in Section 2.2.1, χ_q(t, t^′) and ${\chi }_{\mathit{q}}^{\prime }\left(t,t^{\prime }\right)$ can be explicitly expressed in terms of the retarded Green function given by Eq. (20). Here, the obtained results are shown below; for more detail, consult Ref. [31]:

${\chi }_{\mathit{q}}^{\ast }\left(t,t^{\prime }\right)=\frac{4\pi }{V}\sum _{\alpha {\alpha }^{\prime }\beta {\beta }^{\prime }}{N}_{\mathit{q}\alpha }^{L\ast }\left(t\right)V_{\mathit{q}\mathit{\alpha \beta }}^L\left(t\right)G_{\mathit{q}\beta {\beta }^{\prime }}^R\left(t,t^{\prime }\right)V_{\mathit{q}{\beta }^{\prime }{\alpha }^{\prime }}^{L†}\left(t^{\prime }\right)N_{\mathit{q}{\alpha }^{\prime }}^L\left(t^{\prime }\right),$E23

where $N_{\mathit{q}\alpha }^L={\sum }_{\mathit{k}b}{U}_{\mathit{q}\alpha }^{L†}\left(\mathit{k}\mathit{bb}\right)$, and this is equivalent to a normalization constant of the left vector $U_{\mathit{q}\alpha }^{L†}$:

${\chi }_{\mathit{q}}^{\prime }\left(t,t^{\prime }\right)=\frac{4\pi }{V}{\left|g_{\mathit{q}}^{\prime }\right|}^2\left[{{\overline{D}}^{\prime }}_{\mathit{q}}^R\left(t,t^{\prime }\right)+{\left[{{\overline{D}}^{\prime }}_{-\mathit{q}}^R\left(t,t^{\prime }\right)\right]}^{\ast }\right],$E24

where $g_{\mathit{q}}^{\prime }$ is a constant in proportion to (g_cq + g_vq)/2 and

${{\overline{D}}^{\prime }}_{\mathit{q}}^R\left(t,t^{\prime }\right)=\sum _{\beta {\beta }^{\prime }}{V}_{\mathit{q}{\alpha }_2\beta }^L\left(t\right)G_{\mathit{q}\beta {\beta }^{\prime }}^R\left(t,t^{\prime }\right)V_{\mathit{q}{\beta }^{\prime }{\alpha }_2}^{L†}\left(t^{\prime }\right).$E25

Finally, the TFR dynamics caused by the CP generation is mentioned based on the PQ picture. As shown in Figure 2, the LO-phonon state α₂ is embedded in the quasiboson state β, and the effective coupling between both states induces the formation of transient PQ FR state. This composite state is deexcited into the PQ ground state via two paths: one is the transient photoemission from α₂, and the other is from β. It is likely that these two paths interfere to give rise to asymmetry in spectra. It is remarked that the contribution from the plasmon-like mode α₁ is omitted because of a negligibly smaller effect on the TFR.

Figure 2.

Schematic diagram of the TFR dynamics based on the PQ picture, where the LO-phonon state α₂ is embedded in the quasiboson state β. The PQ FR state composed of α₂ and β is deexcited by induced photoemission process. For more detail, consult the text (from Ref. [31] with partial modification).

3. Results and discussion

3.1. DFR in the photodressed exciton

For the calculations of DFR spectra, the semiconductor SLs of GaAs/Ga_0.75Al_0.25As are employed with 35/11 monolayers (ML) for the well and barrier thickness, where 1 ML = 2.83 Å. Here, 14 photon sidebands of $\overline{\mu }=\left[1,1,-3\sim 3\right]$ and [2, 1, −3 ∼ 3] are incorporated by setting ω to 91 meV; this equals to the difference between the centers of the joint minibands of (1,1) and (2,1). Other photon sidebands are neglected for the sake of simplicity.

First of all, the calculated quasienergy bands {$\mathcal{E}$_μ} as a function of F_ac are shown in Figure 3 to illustrate the effect of ac-Zener coupling. The two photon sidebands labeled by μ₁ = [1, 1, 0, k] and μ₂ = [2, 1, −1, k] are mixed by the coupling induced by the driving laser F(t). With the increase of F_ac, the quasienergy bands are branched into two distinct photon sidebands, termed as the upper sideband μ₊ and the lower sideband μ₋, where both labels of μ₁ and μ₂ are no longer good quantum numbers, aside from k. It is noted that both of μ₊ and μ₋ form dynamic localization showing band collapse around two points F_ac = F_DL1 ≡ 170 kV/cm and F_DL2 ≡ 395 kV/cm. Figure 4 shows the absorption spectra α(ω_p; ω) obtained by solving Eq. (9) in the range of F_ac from 10 to 450 kV/cm. Asymmetric spectral profiles characteristic of DFR are discerned at the arrowed positions of ω_p when F_ac≥ 150 kV/cm, where all peaks are followed by dips. These peaks are located just below the upper sideband μ₊, thereby being blue shifted. Consulting Figure 1, the DFR is dominantly formed by the interaction between one open channel μ₋ and one closed channel μ₊.

Figure 3.

The quasienergy ℰ_μ as a function of F_ac. ω is set to 91 meV. The vertical double arrows represent the original SL miniband widths corresponding to the photon sidebands of μ₁ and μ₂ (from Ref. [16] with partial modification).

Figure 4.

Absorption spectra α(ω_p; ω) as a function of ω_p for F_ac=50–450 (kV/cm) with ω= 91 meV. A series of the arrowed spectral profiles are examined exclusively in the text. The quasienergies shown in Figure 3 are also plotted (dotted lines) (from Ref. [16] with partial modification).

To deepen the understanding of the DFR exciton, its characteristic quantities determining the spectral profiles are extracted from α(ε) ≡ α(ω_p; ω) arrowed in Figure 4 by being fitted to Fano’s Formula [1]:

$\alpha \left(\epsilon \right)={\alpha }_0\frac{{\left(\epsilon +q^{\left(F\right)}\right)}^2}{{\epsilon }^2+1},$E26

in the vicinity of an excitonic resonance quasienergy $\mathcal{E}$_ex, where ε = 2(ω_p − $\mathcal{E}$_ex)/Γ with the spectral width Γ and the asymmetry parameter (Fano’s q-parameter) q^(F) < 0. Figure 5(a) shows the evaluated values of ∣1/q^(F)∣ and Γ as a function of F_ac, while Figure 5(b) shows the peak intensity α(0) = α₀[q^(F)]² ≡ α_max and background spectra α(±∞) = α₀ as a function of F_ac. It is seen that these functions are affected pronouncedly by F_ac; in particular, extrema are formed around F_ac = F_DL1. It is remarked that with the decrease in ∣1/q^(F)∣ and Γ, the DFR state becomes a pure bound state. In addition, there still exist faint extrema around F_ac = F_DL2 in the concerned quantities except Γ. Therefore, the DL is considered to fulfill a special role of the quantum control of photodressed excitonic states.

Figure 5.

The DFR-related quantities as a function of F_ac with the fixed value of ω = 91 meV. The calculated results represented by the filled symbols are connected by the solid lines in order to aid the presentation. (a) ∣1/q^(F)∣ and Γ and (b) α₀ and α_max (from Ref. [16] with partial modification).

For the purpose of confirming such an effect of DL and the pronounced F_ac dependence of related quantities on the excitonic DFR, one evaluates the transition probability between the photon sidebands of μ₁ and μ₂ due to the ac-Zener coupling; this value is represented as M(F_ac) as a function of F_ac. This corresponds to the degree of mixing between these two photon sidebands. M(F_ac) is readily obtained by solving the associated coupled equations between μ₁ and μ₂ in an approximate manner of neglecting contributions from all other photon sidebands [16]. Given Δε and v as the difference of ac-Zener-free quasienergies between μ₁ and μ₂, and the matrix element of the ac-Zener coupling between them, respectively, M(F_ac) is provided as

$M\left(F_{\mathit{ac}}\right)={\left[sin\left(\phi /2\right)\right]}^2=\frac{1}{2}\left(1-\frac{1}{\sqrt{1+z^2}}\right),$E27

where z ≡ tan φ = 2∣v∣/∣Δε∣. With x = F_acd/ω, v and Δε are evaluated as v ∝ x and Δε ∝ cos(kd)J₀(x), respectively; see Section 1. Thus, one has z = 2x/ηJ₀(x) where η is a proportional constant between Δε and v. According to Eq. (27), for finite values of η, with the increase of F_ac, M(F_ac) increases from 0 to 1/2 in an oscillating manner; for more detail of the shape of M(F_ac) for several values of η, consult Ref. [16].

The alteration pattern of M(F_ac) with respect to F_ac looks somewhat similar to the shapes of the DFR-related functions shown in Figure 5. In particular, it is noted that M(F_ac) has extrema at zeros of J₀(x), which just correspond to DL concerned here; that is, M(F_ac) shows extrema at F_ac = F_DL1 and F_DL2. In fact, M(F_ac) shows a clear extremum at F_ac = F_DL1, while the second extremum at F_ac = F_DL2 is not obviously discernible. This is understood by the behavior that the oscillating component incorporated in J₀(x) is overwhelmed by the ac-Zener coupling v for large x. Therefore, it is concluded that the characteristic F_ac dependence of the functions of ∣1/q^(F)∣, Γ, α_max, and α₀ is attributed to the competition between the ac-Zener effect and the band width of the free electron-hole pair states in the vicinity of the DL positions.

Finally, one mentions in brief the ω dependence of the physical quantities ∣1/q^(F)∣ and Γ at F_ac=180 kV/cm in the vicinity of F_ac = F_DL1. As shown in Figure 6(a), ∣1/q^(F)∣ decreases sharply with the increase in ω, while Γ is maximized around ω = 91 meV at which the centers of two photon sidebands μ₁ and μ₂ coincide. The tendency of ∣1/q^(F)∣ is in harmony with the ω dependence of the ratio of d_c to d_o, namely, r_d = d_c/d_o, as shown in Figure 6(b), where d_c and d_o represent a dipole-transition matrix from the ground state to the closed channel μ₊ and that to the open channel μ₋, respectively. Actually, r_d is in proportion to q^(F) [16]. Such alteration of r_d is interpreted on the basis of the anticrossing formation of photon sidebands of μ₊ and μ₋ due to the Autler-Townes effect, though not discussed here; for more detail, consult Ref. [16]. Thus, it seems that comparing Figure 6(a) with Figure 5(a), the q parameter is even more controllable by changing ω than by F_ac.

Figure 6.

The DFR-related quantities as a function of ω with the fixed value of F_ac = 180 meV. The calculated results represented by the filled symbols are connected by the solid lines in order to aid the presentation. (a) ∣1/q^(F)∣ and Γ and (b) ∣r_d∣ (from Ref. [16] with partial modification).

3.2. TFR in the CP generation

For the calculations of TFR spectra of undoped Si and undoped GaAs, the associated materials parameters employed in the present study are shown in Ref. [31], while parameters of a square-shaped pulse laser employed are as follows. For undoped Si and undoped GaAs, detuning with reference to energy band gap Δω=82 and 73 meV, respectively, temporal width τ_L=15 fs, pulse area A_L=0.12π and 0.20π, respectively, and the maximum excited electron density $N_{\mathit{ex}}^0$=6.31 × 10¹⁷ and 5.30 × 10¹⁷ cm³, respectively; by Δω > 0, it is meant that opaque interband transitions with real excited carriers are examined. Further, two time constants of T₁₂ and T_q12 are introduced, which represent phenomenological damping time constants of induced carrier density with isotropic momentum distribution and anisotropic momentum distribution with q, respectively. The temporal region t < T₁₂ is termed as the early-time region during which a great number of carriers still stay in excited states, and the quantum processes govern the CP dynamics; T_q12 is approximately equal to T_q(kbb^′) introduced in Eq. (14). On the other hand, the temporal region t ≳ T₁₂ is termed as the classical region. For the present calculations, T_q12 and T₁₂ are set equal to 20 and 90 fs, respectively. As regards experimental estimates of these time constants for Si, T_q12 and T₁₂ extracted from the CP measurements in Ref. [45] are 16 and 100 fs, respectively, at $N_{\mathit{ex}}^0=4\times {10}^{19}{\mathrm{cm}}^{-3}$.

Transient induced photoemission spectra ${\overline{A}}_{\mathit{q}}\left(t_p,{\omega }_p\right)$ defined in Section 2.2.2 show the change of excited electronic structure due to the pump field at probe time t_p, and this is crucial to understand the TFR accompanied by CP generation. The total retarded longitudinal susceptibility consists of the dynamically screened Coulomb interaction induced by electron and the LO-phonon-induced interaction. That is, ${\tilde{\chi }}_{\mathit{q}}^{\left(t\right)}\left(t_p;{\omega }_p\right)={\tilde{\chi }}_{\mathit{q}}\left(t_p;{\omega }_p\right)+{\tilde{\chi }}_{\mathit{q}}^{\text{'}}\left(t_p;{\omega }_p\right)$, where this is a Fourier transform of Eq. (22) with respect to τ into the ω_p domain. In the small transferred momentum q limit, ${\tilde{\chi }}_{\mathit{q}}\left(t_p;{\omega }_p\right)$ is proportional to |q|², while ${\tilde{\chi }}_{\mathit{q}}^{\text{'}}\left(t_p;{\omega }_p\right)$ is proportional to |q|² for the Fröhlich interaction exclusively for a polar crystal such as GaAs and to |q|⁴ for the deformation potential interaction. This difference is attributed to the fact that the Fröhlich interaction is of long range, and the deformation potential interaction is of short range. It is noted that in a nonpolar crystal such as Si, a dipole transition for lattice absorption vanishes in the limit of q = 0 because of the presence of spatial inversion symmetry [61].

In Figures 7 and 8, ${\overline{A}}_{\mathit{q}}\left(t_p;{\omega }_p\right)$ of Si and GaAs as a function of ω_p is shown, respectively, by solid lines at t_p equal to t₁ ≡ 15 ,  t₂ ≡ 65 and t₃ ≡ 100 fs, where the separate contributions from ${\tilde{\chi }}_{\mathit{q}}\left(t_p;{\omega }_p\right)$ and ${\tilde{\chi }}_{\mathit{q}}^{\text{'}}\left(t_p;{\omega }_p\right)$ are also shown by chain and dashed lines, respectively. ${\tilde{\chi }}_{\mathit{q}}\left(t_p;{\omega }_p\right)$ and ${\tilde{\chi }}_{\mathit{q}}^{\text{'}}\left(t_p;{\omega }_p\right)$ are mostly governed by the plasmon-like mode α₁ and the LO-phonon mode α₂, respectively. In both figures, it is seen that just ${\tilde{\chi }}_{\mathit{q}}^{\text{'}}\left(t_p;{\omega }_p\right)$ contributes to the formation of spectral peaks and becomes dominant over ${\tilde{\chi }}_{\mathit{q}}\left(t_p;{\omega }_p\right)$ in the classical region.

Figure 7(a) shows ${\overline{A}}_{\mathit{q}}\left(t_p;{\omega }_p\right)$ of Si at t_p = t₁ < T_q12, where the obtained continuum spectra are governed by the contribution from ${\tilde{\chi }}_{\mathit{q}}\left(t_p;{\omega }_p\right)$, whereas the contribution from ${\tilde{\chi }}_{\mathit{q}}^{\text{'}}\left(t_p;{\omega }_p\right)$ is negligibly small due to the proportion of it to |q|⁴. In Figure 7(b) at t_p = t₂ with T_q12 < t_p < T₁₂, the contributions from ${\tilde{\chi }}_{\mathit{q}}\left(t_p;{\omega }_p\right)$ are damped to be comparable to those from ${\tilde{\chi }}_{\mathit{q}}^{\text{'}}\left(t_p;{\omega }_p\right)$. It is noted that asymmetric spectra characteristic of FR are manifested with a dip followed by a peak. This is in sharp contrast with a symmetric Lorentzian profile shown in Figure 7(c) at t_p = t₃ > T₁₂. As regards ${\overline{A}}_{\mathit{q}}\left(t_p;{\omega }_p\right)$ of GaAs, it is shown in Figure 8(a) that at t_p = t₁, a pronounced peak due to the α₂ mode, is superimposed with a continuum background composed of ${\tilde{\chi }}_{\mathit{q}}\left(t_p;{\omega }_p\right)$ and ${\tilde{\chi }}_{\mathit{q}}^{\text{'}}\left(t_p;{\omega }_p\right)$ with comparable order, since both are in proportion to |q|². The spectra at t_p = t₂ shown in Figure 8(b) are dominated by ${\tilde{\chi }}_{\mathit{q}}^{\text{'}}\left(t_p;{\omega }_p\right)$, differing a lot from those shown in Figure 7(b) of Si. The spectra at t_p = t₃ in Figure 8(c) are similar to those in Figure 7(c).

Figure 7.

${\overline{A}}_{\mathit{q}}\left(t_p,{\omega }_p\right)$ of undoped Si (solid line) as a function of ω_p at t_p equal to (a) 15 fs, (b) 65 fs, and (c) 100 fs. Separate contributions to the spectra from ${\tilde{\chi }}_{\mathit{q}}\left(t_p;{\omega }_p\right)$ and ${\tilde{\chi }}_{\mathit{q}}^{\text{'}}\left(t_p;{\omega }_p\right)$ are also shown by chain and dashed lines, respectively. ${\overline{A}}_{\mathit{q}}\left(t_p;\omega \right)$ is reckoned from structureless background due to electron-hole continuum states β that are almost constant in the ω_p region concerned. The widths of the spectral peaks are determined by a phenomenological damping constant T_ph of LO-phonon due to lattice anharmonicity: 2/T_ph = 0.27 meV (from Ref. [31] with partial modification).

Figure 8.

The same as Figure 7 but for undoped GaAs (from Ref. [31] with partial modification).

The origin of the manifestation of TFR in Si shown in Figure 7(b) is examined below. The principal difference between Si and GaAs observed here is attributed just to the effective coupling M_qβ between quasiboson and LO-phonon aside from less significant difference in other material parameters; this appears in the matrix h_q introduced in Eq. (16), and the approximation of $M_{\mathit{q}}\approx M_{\mathit{q}}^{\text{'}}$ is employed here. The primitive coupling constant g_bq incorporated in M_qβ consists of $g_{b\mathit{q}}^D$ and $g_{b\mathit{q}}^F$ representing the coupling constants due to a phenomenological LO-phonon deformation potential interaction and the Fröhlich interaction, respectively, that is, $g_{b\mathit{q}}=g_{b\mathit{q}}^D+g_{b\mathit{q}}^F$. Here, $g_{b\mathit{q}}^D$ is real and approximately independent of q, while $g_{b\mathit{q}}^F$ is pure imaginary and $\mid g_{b\mathit{q}}^F\mid \propto {\left|\mathit{q}\right|}^{-1}$ [61]. In a nonpolar crystal such as Si, $g_{b\mathit{q}}=g_{b\mathit{q}}^D$, whereas in a polar or partially ionic crystal such as GaAs, $g_{b\mathit{q}}^F$ is much dominant to $g_{b\mathit{q}}^D$, namely, $g_{b\mathit{q}}\approx g_{b\mathit{q}}^F$. Actually, the phase of M_qβ is almost determined by that of g_bq, since a residual factor defining M_qβ is almost considered real. Thus, M_qβ is a complex number given by M_qβ = ∣M_qβ∣e^iφ_qβ in general; φ_qβ = 0 , π for Si, while φ_qβ =  ± π/2 for GaAs.

Next, discussion is made on how such difference of M_qβ affects the spectral profile of ${\overline{A}}_{\mathit{q}}\left(t_p;{\omega }_p\right)$ based on the PQ picture depicted in Figure 2. It is seen that there are two transition paths for the process: one is a direct path mediated by an optical transition matrix $D_{\mathit{q}{\alpha }_2}^{\left(r\right)}$ from LO-phonon state α₂ to the PQ ground state, and the other is a two-step resonant path mediated by M_qβ from α₂ to quasiboson state β, followed by a deexcited process mediated by an optical transition matrix $D_{\mathit{q}{\alpha }_2}^{\left(c\right)}$ from β to the PQ ground state. Accordingly, owing to Shore’s model [62], the induced photoemission spectra in the proximity of ${\omega }_p\approx {\omega }_{\mathit{q}}^{\left(\mathit{LO}\right)}$ is read as

${\overline{A}}_{\mathit{q}}\left(t_p;{\omega }_p\right)\approx {\mathcal{C}}_{\mathit{q}\beta }+\frac{{\mathcal{A}}_{\mathit{q}{\alpha }_2}\left({\omega }_p-{\omega }_{\mathit{q}}^{\left(\mathit{LO}\right)}\right)+{\mathcal{B}}_{\mathit{q}{\alpha }_2}{\mathrm{\Gamma }}_{\mathit{q}{\alpha }_2}/2}{{\left({\omega }_p-{\omega }_{\mathit{q}}^{\left(\mathit{LO}\right)}\right)}^2+{\left({\mathrm{\Gamma }}_{\mathit{q}{\alpha }_2}/2\right)}^2},$E28

where a set of Shore’s spectral parameters of ${\mathcal{A}}_{\mathit{q}{\alpha }_2}$, $\mathcal{B}$_qα2, and ${\mathcal{C}}_{\mathit{q}\beta }$ are provided by

$\left\{\begin{array}{l}{\mathcal{A}}_{\mathit{q}{\alpha }_2}=2\mid D_{\mathit{q}\beta }^{\left(c\right)}\mid \mid D_{\mathit{q}{\alpha }_2}^{\left(r\right)}\mid \mid M_{\mathit{q}\beta }\mid cos{\phi }_{\mathit{q}\beta }\hfill \\ {\mathcal{B}}_{\mathit{q}{\alpha }_2}=-2\mid D_{\mathit{q}\beta }^{\left(c\right)}\mid \mid D_{\mathit{q}{\alpha }_2}^{\left(r\right)}\mid \mid M_{\mathit{q}\beta }\mid sin{\phi }_{\mathit{q}\beta }+{\left|D_{\mathit{q}{\alpha }_2}^{\left(r\right)}\right|}^2{\left|M_{\mathit{q}\beta }\right|}^2/\left({\mathrm{\Gamma }}_{\mathit{q}{\alpha }_2}/2\right)\hfill \\ {\mathcal{C}}_{\mathit{q}\beta }={\left|D_{\mathit{q}\beta }^{\left(c\right)}\right|}^2\hfill \end{array}\right.$E29

and the natural spectral width is represented by Γ_qα2 = 2πρ_qα2|M_qα2|²; ρ_qα2 and M_qα2 are the density of state of quasiboson and the coupling matrix at ${\mathcal{E}}_{\mathit{q}\beta }={\omega }_{\mathit{q}}^{\left(\mathit{LO}\right)}$, respectively. The associated Fano’s q parameter is determined in terms of Shore’s parameters as $q_{\mathit{q}{\alpha }_2}\left(t_p\right)=r_{\mathit{q}{\alpha }_2}\left(t_p\right)+{\sigma }_{\mathit{q}{\alpha }_2}\left(t_p\right)\sqrt{{\left[r_{\mathit{q}{\alpha }_2}\left(t_p\right)\right]}^2+1}$ with $r_{\mathit{q}{\alpha }_2}\left(t_p\right)={\mathcal{B}}_{\mathit{q}{\alpha }_2}/{\mathcal{A}}_{\mathit{q}{\alpha }_2}$ and ${\sigma }_{\mathit{q}{\alpha }_2}\left(t_p\right)={\mathcal{A}}_{\mathit{q}{\alpha }_2}/\mid {\mathcal{A}}_{\mathit{q}{\alpha }_2}\mid .$

An asymmetric spectral profile is exclusively determined by ${\mathcal{A}}_{\mathit{q}{\alpha }_2}$. It is seen that ${\mathcal{A}}_{\mathit{q}{\alpha }_2}\left(t_p\right)$ vanishes for φ_qβ =  ± π/2 and ${\overline{A}}_{\mathit{q}}\left(t_p;{\omega }_p\right)$ becomes of symmetric shape with ∣q_qα2(t_p)∣ infinite. The spectral profile of GaAs shown in Figure 8(b) corresponds to this case. For φ_qβ ≠  ± π/2, both ${\mathcal{A}}_{\mathit{q}{\alpha }_2}\left(t_p\right)$ and $\mathcal{B}$_qα2(t_p) are finite, and ${\overline{A}}_{\mathit{q}}\left(t_p;\omega \right)$ becomes of asymmetric shape with ∣q_qα2(t_p)∣ finite. The spectral profile of Si shown in Figure 7(b) corresponds to this case, where φ_qβ ≈ 0 , π. For Figures 7(c) and 8(c), since $D_{\mathit{q}\overline{\alpha }}^{\left(c\right)}$ and ∣M_qβ∣ become negligibly small, ${\overline{A}}_{\mathit{q}}\left(t_p;\omega \right)$ is governed by the second term of the expression of $\mathcal{B}$_qα2(t_p), and this becomes symmetric with Γ_qα2 ≈ 0. To conclude, the effective coupling M_qβ around ${\mathcal{E}}_{\mathit{q}\beta }\approx {\omega }_{\mathit{q}}^{\left(\mathit{LO}\right)}$ plays the crucial role of the manifestation of TFR, and the asymmetry of profile is mostly determined by φ_qβ as long as ∣M_qβ∣ is still large.

Finally, the manifestation of TFR of Si is discussed from the viewpoint of the allocation of time constants T_q12 and T₁₂, where one sets T_q12 < T₁₂. This is an important issue for deepening the understanding of TFR. As shown in Figure 7(b), in the region of T_q12 ≲ t_p < T₁₂, the asymmetric spectral profile bursts into view from the structureless continuum ${\tilde{\chi }}_{\mathit{q}}\left(t_p;\omega \right)$. Actually, in the early-time region of t_p < T₁₂, the excited carrier density is still populated enough around the energy region of ${\mathcal{E}}_{\mathit{q}\beta }\approx {\omega }_{\mathit{q}}^{\left(\mathit{LO}\right)}$ to couple strongly with LO-phonon, while the effect of ${\tilde{\chi }}_{\mathit{q}}\left(t_p;\omega \right)$ is much suppressed in the region of T_q12 ≲ t_p. As regards a different allocation of these time constants, for instance, T_q12 ∼ T₁₂, the TFR profile is no longer observed in the region of t_p < T₁₂, since this is covered with still dominant contributions from ${\tilde{\chi }}_{\mathit{q}}\left(t_p;\omega \right)$, and the effect of M_qβ becomes too small to cause TFR in the region of t_p ≈ T₁₂. Therefore, the allocation of time constants such as T_q12 < T₁₂ is a necessary condition for realizing the TFR of Si in ${\overline{A}}_{\mathit{q}}\left(t_p;\omega \right)$; otherwise this never appears.

4. Conclusion

Transient and optically nonlinear FR in condensed matter is examined here, which differs from conventional FR processes caused by a weak external perturbation in a stationary system. In particular, the following two FR processes are discussed: one is the DFR of Floquet exciton realized in semiconductor superlattices driven by a strong cw laser, and the other is the TFR accompanied by the CP generated by an ultrashort pulse laser in bulk crystals of undoped Si and undoped GaAs.

It is shown that the physical quantities relevant to the DFR spectra can be controlled by modulating F_ac and ω. In particular, the quantities as a function of F_ac take the extrema due to the ac-Zener coupling between the photon sidebands of μ₁ and μ₂, when F_ac is suitably adjusted to satisfy the DL condition. Further, the strong ω dependence is explained on the basis of the Autler-Townes effect forming the anticrossing between these two photon sidebands. It is remarked that the spectral width shown in Figures 5 and 6 seems too small to be confirmed by experiments. Actually, in the present calculations, the Coulomb many-body effect is neglected. At least at the Hartree-Fock level, the vertex correction to the Rabi energy would make the net ac-Zener coupling stronger to result in such a great DFR width that experimental measurement would be accessible.

As regards the TFR spectra, the PQ model succeeds in demonstrating the appearance of asymmetric spectral profile in Si in a flash, whereas the profile observed in GaAs remains symmetric; the obtained results are in harmony with the existing experimental ones [45]. The difference between Si and GaAs is attributed to the phase factor of the effective coupling M_qβ(t_p). To conclude, it is found that in order to realize the TFR in the CP dynamics, the following conditions are to be fulfilled simultaneously. First, the coupling of an LO-phonon with an electron-hole continuum is conducted by the LO-phonon deformation potential interaction rather than by the Fröhlich interaction. Second, photoexcited carriers are populated enough around the energy region ${\mathcal{E}}_{\mathit{q}\beta }\approx {\omega }_{\mathit{q}}^{\left(\mathit{LO}\right)}$ in the early-time region T_q12 < t_p < T₁₂ with T_q12 ≪ T₁₂.

Acknowledgments

This work was supported by JSPS KAKENHI Grants No. JP21104504, JP23540360, and JP15K05121.